Ab initio characterization of the quantum linear-zigzag transition using density matrix renormalization group calculations

Ions of the same charge inside confining potentials can form crystalline structures which can be controlled by means of the ion density and of the external trap parameters. In particular, a linear chain of trapped ions exhibits a transition to a zigzag equilibrium configuration, which is controlled by the strength of the transverse confinement. Studying this phase transition in the quantum regime is a challenging problem, even when employing numerical methods to simulate microscopically quantum many-body systems. Here we present a compact analytical treatment to map the original long-range problem into a short-range quantum field theory on a lattice. We provide a complete numerical architecture, based on the density matrix renormalization group, to address the effective quantum ${\varphi }^4$ model. This technique is instrumental in giving a complete characterization of the phase diagram, as well as pinpointing the universality class of the criticality.

Figure 1

Linear-zigzag instability in a chain of interacting atoms: (left) linear and (right) zigzag configurations. The transition is either controlled by changing the longitudinal interparticle spacing $a$ or the frequency ${\omega }_T$ of the transverse harmonic confinement [3, 26]. The displacement of a particle from the axis $y=0$ plays the role of local order parameter, discriminating between the disordered (linear) phase and the ordered (zigzag) phase.

Figure 2

Probability densities $|{\psi }_q\left(\tilde{y}\right){|}^2$ as a function of $\tilde{y}$ for the lowest five quantum energy eigenstates of the local Hamiltonian of Eq. (19): $q=1$ (red line), $q=2$ (green line), $q=3$ (blue line), $q=4$ (yellow line), and $q=5$ (gray line). For easy reference we also plot the double well potential profile (black thick line) of the Hamiltonian, in arbitrary energy scale units.

Figure 3

Probability distribution $p\left(q\right)$, plotted as a function of the local basis number $q$. $p\left(q\right)$ is defined as $p\left(q\right)=\langle q\left|{\varrho }_j\right|q\rangle$, where ${\varrho }_j={\text{Tr}}_{j^{\prime }\ne j}\left\{\right|{\Psi }_{g,\tilde{\omega }}^L\rangle \langle {\Psi }_{g,\tilde{\omega }}^L\left|\right\}$ is the reduced density matrix of an arbitrary site $j$ in the bulk of the many-body ground state $|{\Psi }_{g,\tilde{\omega }}^L\rangle$, simulated by the DMRG. Here the system is in the zigzag phase, but close to critical point ($g=0.03$, ${\tilde{\omega }}^2=1.0$): The first two levels (symmetric and antisymmetric double peaks, respectively) have equal populations. The third level probability is two orders of magnitude smaller, and rapidly decaying.

Figure 4

Zigzag-order parameter ${\xi }_L(g,\tilde{\omega })$ as a function of $1/L$. Here the lattice size $L$ ranges within $100\le L\le 1600$, while other parameters are $g=0.08$ and $d=14$. The points have been numerically evaluated; the lines are a guide to the eyes. Each set of data corresponds to a different value of the square trap frequency ${\tilde{\omega }}^2=1.30,1.32,1.34,...,1.58,1.60$ (from top to bottom). The zigzag order parameter is obtained via DMRG using the square root of structure-factor density of Eq. (21).

Figure 5

Phase diagram of an array of ions (i.e., $\alpha =1$) in the $(g,{\tilde{\omega }}^2)$ parameter space, at the thermodynamical limit. Inset: displacement of the critical square frequency $|\Delta {\tilde{\omega }}_c^2|$ from the classical value, as a function of $g$. Cyan lines represent a power-law fit of the whole data curve using $|\Delta {\tilde{\omega }}_c^2|\left(g\right)=u{g}^v$ via $u$ and $v$ (see text).

Figure 6

Two-point correlations $G\left(\Delta j\right)$ of the displacement $\tilde{y}$ as a function of the distance $\Delta j$ (magenta dots), acquired via numerical ground-state simulation with the DMRG. Data are fitted using $G_{\mathrm{fit}}\left(\Delta j\right)$ in Eq. (26) (black solid line). The green and orange lines show fits via an exponential and a power law, respectively. Inset: fitted anomalous dimension exponents $\eta$ over several points of the critical boundary, characterized by different values of $g$; the dashed line shows the average value $\eta =0.258\pm 0.012$ while the solid line is the value of $1/4$ predicted from the Ising model theory.

Figure 7

Order parameter ${\xi }_L(g,\tilde{\omega })$, obtained via DMRG, as a function of $\tilde{\omega }$, plotted for different system sizes $L=100,120,...,300$, at $g=0.12$. Inset: rescaled data ${\xi }_L({\tilde{\omega }}_c+\Delta {\tilde{\omega }}_c{L}^{{\gamma }_2})L^{{\gamma }_1}$, with ${\gamma }_1=0.127$ and ${\gamma }_2=1.04$, characterizing $f\left(x\right)$ (see text).

Figure 8

Right inset: von Neumann entropy $\mathcal{S}\left({\rho }_{\ell }\right)$ (black dots) of the reduced density matrix ${\rho }_{\ell }$ calculated applying Eq. (28) to the ground-state simulation with DMRG, as a function of the block size $\ell$. The data shows remarkable agreement with the fit using ${\mathcal{S}}_{\text{th}}\left({\rho }_{\ell }\right)$ of Eq. (29) via $c$ and $c^{\prime }$ (dashed yellow line). Main: $c$ values as a function of the system size $L=100,120,...,300$ for different trap frequencies ${\tilde{\omega }}^2=0.383,0.384,...,0.390$ (top to bottom) at $g=0.12$. The magenta data set detects the critical point at ${\tilde{\omega }}_c^2=0.385$, and the corresponding estimated $c$ value (dashed black line) is $c=0.486$. Left inset: estimated central charge $c$ values at various points in the critical boundary, as a function of the parameter $g$. The dashed line corresponds to the average of $c=0.487$, while the solid line is the theoretical prediction $c_{\text{th}}=1/2$.

Figure 9

Original long-range single-phonon (i.e., $q=1$) band profile $\Xi \left[k_1\right]$ of Eq. (A4), for $\alpha =1$ (purple curve). The parabola (dashed line) is the second-order expansion around the soft mode $k=\pi$, the green curve is the dispersion relation for the corresponding model with nearest-neighbor interaction, and matches the original dispersion band up to $\delta k^2$ around $\pi$.

Figure 10

$q=2$ scattering function $\Xi [k_1,k_2,k_3]$ of Eq. (A4), along some high-symmetry lines of the cubic Brillouin zone, of the original long-range model (purple line), encompassing $\Gamma =(0,0,0)$ (the center of the cube), $X=(0,0,\pi )$ (the face center), and $M=(0,\pi ,\pi )$ (the edge center). The dashed line is the second-order expansion around the soft mode point $\vec{\pi }$; the green line is the effective short-range model.